Parents can refer without a medical professional. Here's the link to the referral form:
https://osse.dc.gov/sites/default/files/dc/sites/osse/publication/attachments/Strong%20Start%20DC%20EIP%20Referral%20Form%20English_updated.pdf
Your child would meet the eligibility criteria if he/she has a delay of at least 25% in at least one area of development (communication, fine motor, gross motor cognition, social emotional or adaptive/self help) or a diagnosed condition that lends itself to a developmental delay.